AD-781 499

PROCEDURES FOR DETECTING OUTLYINGOBSERVATIONS IN SAMPLES

Frank E. Grubbs

Ballistic Research LaboratoriesAberdeen Proving Ground, Maryland

April 1974

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Procedures for Detecting Outlying FinalObservations in Samples

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'I .EY *011OS 'coorump. -.s reso &$do st wcosawn -0 ,..ftIT 1W Mec8"afDetection of OutliersDiscrepant ObservationsData ScreeningSignificance Tests for Outliers

to ASSTRacT? ermbwMrvtose otom WWet W"ka~ (tbs)Procedures are given in this report for determining statisticallywhether the highest observation, or the lowest observation, or the0ighest and louest observations, or the two highest observations,r the two lowest observations, or perhaps more of the observationsn the sample may be considered to be outlying observations orfiscrepant values. Statistical tests of significance are useful inI fhis connection either in the absence of assignable physical causesDD, ',O' 1473 ro-Ticmar4Ov NOVAIS OSO.ET UNCLASSIFIED

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or to support a practical judgement that some of the experimentalobservations are aberrant. Both the statistical formulae ardillustrative applications of the procedures to practical exarples;are given, thus representing a rather complete treatment ofsignificance tests for outliers in single univariate s2sples. Thireport has been prepared primarily as a useful guide or as anexpository and tutorial approach to the problem of detectingoutlying observations in much experimental work. We cover onlystatistical tests of significance in this report and appropriateinterpretations which one might draw therefrom.

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UNCLASSIFIEDSECUSTy CLASSFICArIlo Or T#MS oGE VIYwa Dwo Fnf4~w)

PROCEDURES FOR DETECTING OUTLYING OBSERVATIONS IN SAMPLES*

SUMMARY

Procedures are given in this report for determiningstatistically whether the highest observation, or thelowest observation, or the highest and lowest observations,or the two highest observations, or the two lowestobservations, or perhaps more of the observations in thesample may be considered to be outlying observations ordiscrepant values. Statistical tests of significance areuseful in this connection either in the absence of assign-able physical causes or to support a practical judgementthat some of the experimental observations are aberrant.Both the statistical formulae and illustrative applicationsof the procedures to practical examples are given, thusrepresenting a rather complete treatment of significancetests for outliers in single univariate samples. Thisreport has been prepared primarily as a useful guide oras an expository and tutorial approach to the problem ofdetecting outlying observations in much experimental work.We cover only statistical tests of significance in thisreport and appropriate interpretations which one mightdraw therefrom.

*Subs:antially the same material of this report is toappear in a recommended practice of the American Societyfor Testing and Materlals for dealing with outlyingobservations. The author is a member of Committee E-ll onStatistical Methods of ASTK and was encouraged by variousmembers of the Society to prepare much of the materialcovered herein. He gratefully acknowledges their suggestionsand advice.

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TABLE OF CONTENTS

SUMM4ARY ................................ 3

TABLE OF CONTENTS ............................. 5

LIST OF TABLES......................7I . INTRODUCTION .. . .. . ......................... 9

11. GENERAL ........ ... .. ...... ..... 9

III. BASIS C' STATISTICAL CRITERIA FOR OUTLIERS ....... 10

IV. RECOMMENDED CRITERIA FOR SINGLE SAMIPLES.......... 12

V. RECOMMIENDED CRITERION USING INDEPENDENT STANDARDDEVIATION .. *.......*...... . o..*.....e....... 2S

VI. RECOMMENDED CRITERIA FOR KNOWN STANDARD DEVIATION. 28

VII. ADDITIONAL COMMENTS .......................... 29

REFERENCES ... ... ...... . .... . ......... . ... ...... S3DISTRIBUTION LIST ...... o.....................*... 55

Preceflug pop laot

LIST OF TABLES

Page No.

TABLE 1 ............................................... . 31

TABLE 2 ................................................. 35

TABLE 3 . ................................................. 36

TABLE 4 .................. 37

TABLE 5 ............. .. ..... o .............. . ......... .. 41

TABLE 6 ..... *......... . ....................... 42TABLE 7 .. . . . . . . . . . . . . . . . . . . . . ...... 43TABLE 8 ....................................... 43

TABLE 9 .............................. 0.................. 44

TABLE 10 ..... .. ........................ 46

TABLE 11 .................. ................... 46TABLE 1Z .......... ........... ....... 0.................. so

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I. INTRODUCTION

This report deals with the problem of outlyingobservations in samples and how to test the statisticalsignificance of them. An outlying observation, or "outlier",is one that appears to deviate markedly from other membersof the sample in which it occurs. In this connection, thefollowing twc alternatives are of interest:

(a) An outlying observation may be merely an extrememanifestation of the random variability inherent in thedata. If this is true, the values should be retained andprocessed in the same manner as the other observations inthe sample.

(b) On the other hand, an outlying observation may bethe result of gross deviation from prescribed experimentalprocedure or an error in calculating or recording thenumerical value. In such cases, it may be desirable toinstitute an investigation to ascertain the reason for theaberrant value. The observation may even eventually berejected as a result of the investigation, though notnecessarily so. At any rate, in subsequent data analysis theoutlier or outliers will be recognized as probably beingfrom a different process or universe than that of the samplevalues.

(c) It is our purpose here to provide statisticalrules that will lead the experimenter almost unerringly tolook for causes of outliers when they really exist, and henceto decide whether alternative (a) above is not the moreplausible hypothesis to accept, as compared to alternative(b), in order that the most appropriate action in furtherdata analysis may be taken. The procedures covered hereinapply primarily to the simplest kind of experimental data,that is, replicate measurements of some property of agiven material, or observations in a supposedly single randomsample. Nevertheless, the tests suggested do cover a wideenough range of cases in practice to have broad utility.

II. GENERAL

When the experimenter believes that a gross deviationfrom prescribed experimental procedure has taken place, theresultant observation should be discarded, whether or not itagrees with the rest of the data and without recourse tostatistical tests for outliers. If a reliable correctionprocedure, for example, for temperature, is available, theobservation may sometimes be corrected and retained.

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In many cases evidence for deviation from prescribedprocedure will consist primarily of the discordant valueitself. In such cases it is advisable to adopt a cautiousattitude. Use of one of the criteria discussed below willsometimes permit a clear-cut decision to be made. Indoubtful cases the experimenter's judgement will haveconsiderable influence. When the experimenter cannotidentify abnormal conditions, he should at least report thediscordant values and indicate to what extent they have beenused in the analysis of the data.

Thus, for purposes of orientation relative to the over-ill problem of experimentation, our position on the matter,o screening samples for outlying observations is precisely-he following:

Physical Reason Known or Discovered for Outl-tier(s):

(a) Reject observation(s).(b) Correct observation(s) on physical grounds.(c) Reject it (them) and possibly take additional

,bservation(s).

PhysicaZ Reason Unknown - Use Statistical Test:

(a) Reject observation(s).(b) Correct observation(s) statistically.(c) Reject it (them) and possibly take additional

-, ciervation(s).(d) Employ truncated sample theory for censored

observations.

The statistical test: may always be used to support a

judgment that a physical reason does actually exist for anoutlier, or the statistical criterion may be used routinelyas a basis to initiate action to find a physical cause.

III. BASIS OF STATISTICAL CRITERIA FOR OUTLIERS

There are a number of criteria for testing outliers.in all of these, the doubtful observation is included in!he calculation of the numerical value of a sample criterion(or statistic), which is then compared with a critical value

based on the theory of random sampling to determine whetherihe doubtful observation is to be retained or rejected. The

ritical value is that value of the sample criterion whichoUld be exceeded by chance with some specified (small)

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probability on the assumption that all the observationsdid indeed constitute a random sample from a common system ofcauses, a single parent population, distribution or universe.The specified small probability is called the "sigficancelevel' or "percentage point" and can be thought o as the riskof erroneously rejecting a good observation. It becomesclear, therefore, that if there exists a real shift orchange in the value of an observation that arises fromnon-random causes (human error, loss of calibration ofinstrument, change of measuring instrument, or even changeof time of measurements, etc.), then the observed value of thesample criterion used would exceed the "critical value"based on random sampling theory. Tables of critical valuesare usually given for several different significance levels,for example, 5 percent, 1 percent. For statistical testsof outlying observations, it is generally recommended thata low significance level, such as 1 percent, be used andthat significance levels greater than 5 percent would notbe common practice.

Sote Z - In this report, we will usually illustrate theuse of the 5 percent significance level. Proper choice oflevel in probability depends on the particular problem andjust what may be involved, along with the risk that one iswilling to take in rejecting a good observation, that is,if the null-hypothesis stating "all observations in thesample come from the same normal population" may beassumed.

It should be pointed out that almost all criteria foroutliers are based on an assumed underlying normal (Gaussian)population or distribution. When the data are not normallyor approximately normally distributed, the probabilitiesassociated with these tests will be different. Until suchtime as criteria not sensitive to the normality assumptionare developed, the experimenter is cautioned againstinterpreting the probabilities too literally.

Although our primary interest here is that of detectingoutlying observations, we remark that some of the statisticalcriteria presented may also be used to test the hypothesisof normality or that the random sample taken did come froma normal or Gaussian population. The end result is forall practical purposes the same, that is, we really wishto know whether we ought to proceed as if we have in hand asample of homogeneous observations.

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IV. RECOMMENDED CRITERIA FOR SINGLE SAMPLES

Let the sample of n observations be denoted in orderof increasing magnitude by x, _ x2 .x ... Xn . Let×n be the doubtful value, that is the largest value. The

test criterion, Tn, recommended here for a single outlier

is as follows:

T x -n s

where

x = arithmetic average of all n values, ands = estimate of the population standard deviation based

on the sample data, calculated as follows:

* =rExi )2 11/2 Inx =zx) /s = n -1= n(n -1If x, rather than xn is the doubtful value, the

criterion is as follows:

x -1T1=Tj s

The critical values for either case, for the 1 and 5percent levels of significance, are given in Table 1.Tahle 1 and the following tables at the end of the reportgive the "one-sided" significance levels. In many previoustreatments of outliers, the tables listed values ofsignificance levels double those in the present i-eport,since it was considered that the experimenter would testeither the lowest or the highest observation (or both) foritatistical significance, for example. However, to becinsistent with actual practice and in an attempt to avoidfrther misunderstanding single-sided significance levelsare tabulated herein so that both viewpoints can berepresented.

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The hypothesis that we are testing in every case isthat all observations in the sample come from the samenormal population. Let us adopt, for example, asignificance level of 0.05. If we are interested onZy inoutliers that occur on the htgh aide, we should always

use the statistic Tn = (xr, - i0/s and take as critical

value the 0.05 point of Table 1. On the other hand, if weare interested onZy in outliers occurring on the Zow side,

we would always use the statistic T, = (i - xl)/s and again

take as a critical value the 0.05 poinu of Table 1. Suppose,however, that we are interested in outliers occurring oneither side, but do not believe that outliers can occur onboth sides simultaneously. We might, for example, believethat at some time during the experiment something possiblyhappened to cause an extraneous variation on the high sideor on the low side, but that it was very unlikely that twoor more such events could have occurred, one being anextraneous variation on the high side and the other anextraneous variation on the low side. With this point ofview we should use the statistic Tn = (x, - i)/s or the

statistic T, = (i - xl)/s whichever is larger. If inthis instance we use the 0.05 point of Table 1 as ourcritical value, the true significance level would betwice 0.05 or 0.10. If we wish a significance level of0.05 and not 0.10, we must in this case use as a criticalvalue the 0.025 point of Table 1. Similar considerationsapply to the other tests given below.

Example l.As an illustration of the use of Tn and Table I,

consider the following ten observations on breakingstrength (in pounds) of 0.104-in. hard-drawn copper wire:568, 570, 570, 570, 572, 572, 572, 578, 584, 596. Thedoubtful observation is the high value, x10 = 596. Is the

value of 596 significantly high? The mean is i = 575.2and the estimated standard deviation is s f 8.70. We compute

- 596 - 575.2 = 2.39U.70

-oiIFrom Table 1, for n = 10, note that a TI.,, as large as

2.39 would occur b~y chance withi probability less than 0.05.:r. fact, so large a value would occui- y :hancr= not muchmrore often thlan one percent of the time. Thus, the weightof the evidenice is against the doubtful value having comefrom the same population as the others (assuming thepopulation is normally distributed). Investigation of the6.oubtful value is therefore indicated.

An alternativec system, the Dixon criteria, based entirelyon ratios of differences between the2 observations isc escribcd in the literature (7) and may be used in caseswhere it is desirable to avoid calculation of s or wherecquickt _udge menlt is called for. For the Dixon test, th2sar.-ple criterion or statistic chaniges with sample size.Table 2gives the appropriate statistic to calculate andalso gives the critical values of the statistic for the1. 5, and 10 percent. levels of significance.

.7-As an illustration of the use of Dixon'sLcsc, con sider again the observations on breaking strengthgivc~n in 'xample 1, and suppose that a large number ofsu-ch samples had lo be screened quickly for outliers and itw-as iudged :-oo rime-consuring t.o compute s. Table 2I'n-licites use Of

Xn X2

"-us, for n =10,

-C Xc.

Ti1l -,, x 2

':'-r tii ;mc-easurements of breaking strength above,

rl.596 -584 = 0,462M96577

V-:hi~h is a little less than 0,477, the 5 percent critical,,,a1iic for n 10. Under the Dixon criterion, we shouldt 1)r _r _, . c co nc thi cSob se r a t io n a s a n o utli e r a tthe 5 percent level of significance. These resultsil lustrate how, borderlinc cases nay 'oe accepted under one

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test but rejected under another. It should be remembered,however, tLat the T--statistic discussed above is the bestone to use for the single-outlier case, and final statisticalJudgment should be based on it, See Ferguson (8,9).

Further examination of the sample observations onbreaking strength of hand-drawn copper wire indicates thatnone of the other values need testing.

,Vote "- With experience we may usually just look at thesample values to observe if an outlier is present. Ilowever,strictly speaking the statistical test should be applied toall samples to guarantee the significance levels used.Concerning "multiple" tests on a single sample, we commenton this below.

A test equivalent to T (or T,) based on the samplensum of squared dev ations from the mean for all the observa-tioi.-z and the sum cf squared deviations omitting the"outlier- is given by Grubbs (11).

The next type of problem to consider is the case herewc have the possibility of two ouLlying observations, theleast and the greatest observation in a sample. (Theproblem of testing the two highest or the two lowestobservations is considered below.) In testing the least andthe greatest observations simultaneously as probable outliersin c sample, ve use. the ratio of sample range to samplestandard deviatlon test of David, Hartley and Pearson (5).Ihe significance levels for this sample criterion are gpven

ii Tab e 3 Alern tiv lythe largest ivtduais te!-t of

Tietjen and Moore (19) could be used.

,..Z - - here is one rather famous set of observa-tions that a number of writers cn the subject of outlyingobservations have referred to in applying their varioustests for "outliers'. This classic set consists of asample of l1. observations ot the vertical semi-diameters ofVenus made by Lieutcnant Herndon in 1846 (2). In thereduction of the observations, Prof. Pierce assumed twounknown quantities and, found Lhe following residuals whichhave been arranged in ascending ordr, o-' magnitude:

-1.40 Ill. -U.24 - i _', 6. 18 h 0.48

-0.44 -0.22 J.\,J' 0.2(0 0.63-0.30 -0.13 0. i, 1.01

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The deviations - 1.40 and 1.01 appear to be outliers.Here the suspected observations lie at each end of thosample. Much less work has been accomplished for thecase of outliers at both ends of the sample than for thecase of one or more outliers at only one end of thesample. This is not necessarily because the "one-sided" case occurs more frequently in practice butbecause "two-sided" tests are much more difficult to dealwith. For a high and a low outlier in a single sample, weFive two procedures below, the first being a combination oftests, and the second a single test o' Tietjen and Moore(19) which may have nearly optimum properties.

For optimum procedures when there is an independentestimate at hand, s2 of c2 , see (6).

For the observations on the semi-diameter of Venusgiven above, all the information on the measurement erroris contained in the sample of 15 residuals. In caseslike this, where no independent estimate of variance isavailable (that is, we still have the single sample case),a useful statistic is the ratio of the range of theobservations to the sample standard deviation:

x xW n - Xs S

. h i r :

If xn is about as far above the mean, x, as x, is below x,and if w/s exceeds some chosen critical value, then one-. ould conclude that botk. the doubtful values are outliers.If, however, x1 and xn are displaced from the mean by

different amounts, some further test would have to be madeto decide whether to reject as outlying only the lowestvalue or only the highest value or both the lowest andhighest values.

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For this example, the mean of the deviations isX 0.018, the sample standard deviation, s = 0.SS1, and

wis -= 1.01 -1.40) _ 2.41 4. 7= O-4.Sl 4 = 2.41 = . 37A

From Table 3 for n = 15, we see that the value ofw/s = 4.374 falls between the critical values for the 1 a-,dS percent levels, so if the test were being run at the Spercent level of significance, we would conclude that thissample contains one or more outliers. Tht. loi..est measure-ment, -1.40 in., is 1.418 below the sample mean and thehighest measurement, 1.01 in., is 0.992 above the mean.Since these extremes are not symmetric about the mean,either both extremes are outliers, or else only -1.40 is anoutlier. That -1.40 is an outlier can be verified byuse of the T1 statistic. We have

0.018 - (-1.40)_ 2.574T,= (x - Xl)/S - . 40.55

This value is greater than the critical value for the Spercent level, 2.409 from Table 1 , so we reject -1.40.Since we have decided that -1.40 should be rejected, weuse the remaining 14 observations and test the upper extreme1.01, either with the criterion

x -xT = nn

or with Dixon's r2 2. Omitting -1.40 and renumbering the

observations, we compute- 1.67x -- - 0.119, s = 0.401,

and

= 1.0i - 0.119 2.220.401

From Table 1, for n 14, we find that a value as large as2.22 would occur by chance more than 5 percent of the time,

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so we should retain the value 1.01 in further calculations.We next calculate

X 14 - X1 ? 1.01 - 0.48 0.53.2 x1, - x4 1.=+ .4 042

From Table 2 for n 14, we see that the 5 percent criticalvalue for r-. is 0.546. Since our calculated value (0.424)

is less than the critical value, we also retain 1.01 byDixon's test, and no further values would be tested in thissample.

. - It should be noted that in a mLltiplicity of testsof this kind, the final over-all significance level will besomewhat less than that used in the individual tests, as weare offering more than one chance of accepting the sample asone produced by a random operation. It is not our purposehere to cover the theory of multiple tests.

For suspected observations on both the high and lowsides in the sample, and to deal with the si-uation invhich some of k > 2 suspected "outliers" are larger andsome smaller than the remaining values in the sample,Tietjen and Moore (19) suggesting the following statistic.Let the sample values be x1 , x2, X 3 ... Xn and compute

the sample mean, x. Then compute the n absolute residuals

rl = 'xl _-x r2 = x2 -~ r = Ix - xl1 1 1, n ... r n

':ow relabel the original observations xl, x2 , ..., xn as

z's in such a manner that z i is that x whose ri is the

ith largest absolute residual above. This now means thatz I is that observation x which is closest to the mean and

that z is the observation x which is farthest from thenmean. The Tietjen-Moore statistic for testing thesignificance of the k largest residuals is then

n-kz (z i _ k ) 2

E i=lk =n(z 2

i=l

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In-k

where Zk E [ zi/(n-k)

is the mean of the (n-k) least extreme observations and zis the mean of the full sample.

Applying this test to the above data, we find that thetotal sum of squares of deviations for the entire sample is4.24964. Omitting -1.40 and 1.01, the suspected twooutliers, wa find that the sum of squares of deviations forthe reduced sample of 13 observations is 1.24089. ThenE, = 1.24089/4.24964 = .292, and by using Table 12, we find

that this observed E2 is slightly smaller than the 5%

critical value of .317, so that the E2 tcst would reject

both of the observations, -1.40 and 1.01. We would probablytake this latter recommendation, since the level of ]significance for the E, test is precisely .05 whereas that A

for the double application of a test for a single outliercannot be guaranteed to be smaller than 1 - (.95)2 = .0975.

The tables of percentage points Ek were computed by

Monte Carlo methods on a high-speed electronic calculator.

We next turn to the case where we may have the twolargest or the two smallest observations as probable outliers.Here, we employ a test provided by Grubbs (10), (11) whichiq based on the ratio of the sample sum of squares when the,.-o doubtful values are omitted to the sample sum :f squareswhen the two doubtful values are included. If simplicityin calculation is the prime requirement, then the Dixontype of test (actually ommitting one observation in the sample)

might be used for this case. In illustrating the testprocedure, we give the following Examples 4 and 5.

Examr;Ze 4 - In a comparison of strength of variousplastic materials, one characteristic studied was thepercentage elongation at break. Before comparison of theaverage elongation of the several materials, it was desirableto isolate for further study any pieces of a given materialwhich gave very small elongation at breakage compared withthe rest of the pieces in the sample. In this example, onemight have primary interest only in outliers to the left ofthe mean for study, since very high readings indicateexceeding plasticity, a desirable characteristic.

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Ten measurements of percentage elongation at breakmade on material No. 23 follow: 3.73, 3.59, 3.94, 4.13,3.04, 2.22, 3.23, 4.05, 4.11, and 2.02. Arranged inascending order of magnitude, these measurements are:2.L2, 2.22, 3,04, 3.23, 3.59, 3.73, 3.94, 4.05, ' .11, 4.13.The questionable readings are the two lowest, 2.02 and2.22. We car. test these two low readings simultaneouslybv using the following criterion of Table 4:

s 21,2I --

For the above measurc-ments:

n n~x~ -S2 = x ( i _ ) =i=l

n

I~q21".3594) - (34,06)" = 5.351,

n nand (n - 2) . - ( 1X )2

Pi= 3 i= (x. - X, .)7 = -i=3 n

8(112.3506) - (29.82)--

9.5724 1197

n

(where X1 2 = xi/(n - 2)]i=3

1%c find:

$21,2 _ 1.197- 2

0- .. 224

Yrom Table 4 for n = 10, the 5 percent significance level

for K /S is 0.2305. Since the calculated value is less

than t e critical value, we should conclude that hoth 2.02.nd 2.22 are outliers. In a situation such as the one

describet! in this example, wbere the outliers are to

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m,,

" " 'i i i l i li i Iii I~i I I i MAN|

be isolated for further analysis, a significance levelas high as 5 percent or perhaps even 10 percent wouldprobably be used in order to get a reasonable size ofsample for additional study. The problem may really be oneof economics, and we use probability as a sensible basisfor action.

ExwpZe 6- The following ranges (horizontal distances inyards from gu- muzzle to point of impact of a projectile)were obtained in firings from a weapon at a constant angleof elevation and at the same weight of charge of propellantpowder:

4782 44204838 48034765 47304549 4"33

It is desired to make a judgment on whether theprojectiles exhibit uniformity in ballistic behavior orif some of the ranges are inconsistent with t.,e others. Thedoubtful values are the two smallest ranges, 4420 and 4S49.For testing these two suspected outliers, the statisticS2 /S 2 of Table 4 is probably the best to use.

1,2

No* 4 - Kudo (15) indicates that if the two outliers aredue to a shift in location or level, as compared to thescale s, then the optimum sample criterion for testingshould he of the type:min(2 x - i x.)/'s = (2Ix x - x2 )/s in our Example S.

The distances arranged in increasing order ofmagnitude are:

4420 47824549 48034730 48334765 4838

The value of S2 is 158,592. Omission of the two shortest

ranges, 4420 and 4549, and recalculation, gives S2

1 2

equal to 8590.8. Thus,

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2

1.2 8590,8 0.054

=- " ~Twhich is significant at 0.01 level (See Table 4). it isthus highly unlikely that the two shortest ranges (occurringactually from excessive yaw) could have come from the samepopulation as that represented by the other six ranges. Itshc.ld be noted that the critical values in Table 4 for the1 percent level of significance are smaller than those forthe 5 percent level.. So for this particular test, thecalculated value is significant if it is less than thechosen critical value.

Fy Monte Carlo methods using an electronic calculator,Tietjen and Moore (19) have recently extended the tables ofpercentage points for the two highest or the two lowestobstiv Lions to k > 2 highest or lowest sample values. Their

n-kresults are given in Table 11 where 1 (x i x)

n n-k(x i - x)2 and x E x /(n-k). Note that theiri=I i

L- equals our or S2 IS4. The columns headed with

an * in Tables 11 and 12 indicate agreement with exact valuescalculated by Grubbs (1950). These new tables may be usedto advantage in many practical problems of interest.

If simplicity in calculation is desired or if alarge number of samples must be examined individually foroutliers, the question-.1ble observations may be tested withthe application of Dixon's criteria. Disregarding thelowest . "nge, 4420 we test if the next lowest range, 4549is outly.ng With n = 7, we see from Table 2 that r10 is

the appropriate statistic. Renumbering the ranges as

x! to x7 , beginning with 4549, we find:

x - x 4730 - 4549 181 0.626r X, X ) &3 4549 "

which is only a little less than the 1 percent criticalvalue, 0.637, for n - 7. So, if the test is being conductedat any significance level greater than a I percent level, we

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I

would conclude that 4549 is an outlier. Since the lowestof the original set of ranges, 4420, is even more outlyingthan the one we have just tested, it can be classified asan outlier without further testing. We note here, however,that this test did not use all of the sample observations.

Rejection of Several Outliers - So far we havediscussed procedures for detecting one or two outliers inthe same sample, but these techniques are not generallyrecommended for repeated rejection, since if severaloutliers are present in the sample the detection of one ortwl spurious values may be "masked" by the presence of otheranc.,alous observations. Outlying observations occur dueto a shift in level (or mean), or a change in scale (thatis, change in variance of the observations), or both.Ferguson (8,9) has studied the power of the variousrejection rules relative to changes in level or scale. Forseveral outliers and repeated rejection of observations,Ferguson points out that the sample coefficient of skewness,

n111U V/ n r (x i - x) 3/[(n 1) )3/S ]

i=l

nTT (xi - X) 3! (x X) 2 3/2i~l

should be used for "one-sided" tests (change in level ofseveral observations in the same direction), and the sanplecoefficient of kurtosis,

nb2= n (xi -)2S

]

nn 7 (xi - x) /[Z(x i - X)2].i=l

is recommended for "two-sided" tests (change in level tohigher and lower values) and also for changes in scale(variance)(see Note 5). In applying the above tests, the

71_ or the b2 , C0 both, are computed and if their observed

23

values exceed those for significance levels given 1r thefollowing tables, then the observation farthest from themean is rejected and the same procedure repeated untilno further sample values are Judged as outliers. (As is

well known v'9 nnd b2 are also used as tests of normality.)

"" . -1n the above equations for 'F1 and bn, s is

defined as used in this standard:

nS = F (Xi - x)2/(n - 1)

i=l

The significance levels in the following tables for

sample sizes of 5, 10, 15 and 20(and 25 for b2 ) were

obtained by Ferguson on an IBM 704 computer using asampling experiment or "Monte Carlo" procedure. Thesignificance levels for the other sample sizes are fromE. S. Pearson, "Table of Percentage Points of

' 1 and b- in Normal Samples; a Rounding off"',.,*..-r';.,

Vol 52; 1965, pp 282 - 285,

SIGNIFICANCE LEVELS FOR ,Fb

SignificanceLevel. % _

5- l 0a 15al 20a1 2 1" 0 1 i -n 1 n' 60i1 1 21 05.41.3 1. 1201 11.0610.98 0.92 0.87 I0.7910.7L 1.05 0.9210.841079 07110.66 0.62 0.59I10.53i.9

SIGNIFICANCE LEVELS FOR b2[Si~n if ican ce

L __5 aT 1 a 1 i 5a] 2 0 a 2 5 a 50 75 100

1 3.1114.83j15.08 5.23 5.0488114.59i4.39

5 2.8913.8514.07 4.154.03.9913.!7f3.77jaThese values were obtained by Ferguson, using a Monte Carlo

procedure. For n = 25, Ferguson's Monte Carlo vaLes ofb2 agree with Pearson's computed values.

24

IL __ _

IThe 'F1 and b:, statistics have the optimum property

of being "locally" best against one-sided and two-sided

alternatives, respectively. The ,'F test is good for up

to 50 percent spurious observations in the sample for theone-sided case and the b, test is optimu,: in the vwo--sidedalternatives case for up to 21 percent "contaminaton" ofsample values. For only one or two outliers the samplestatistics of the previous paragraphs are recormiended, anOFerguson (8) discusses in detail their optimum properties of-o':t ', ol,, one or two outliers.

Instead of the morc complicated b1 and r- statistics,

one can of course use the Tietjen and Moore Tables 11 and12 included herewith for sample sizes and percentage pointsgiven.

V. RECOUMENDED URITERION US IN(" INDEPENDENT STANDARD DEVIATION

Suppose that an independent estimate of the standarddeviation is available from previous data. This estimatenay be from a single sample of previous similar data or maybe the result of combining estimates from several suchprevious sets of data. In any event, each estimate is saidto hav_ dcgrccs of frccdom equal to one less than the samplesize that it is based on. The proper combined estimate is aweighted average of the several values of s "', the weightsbeing proportional to the respective degrees of freedom.The total degrees of freedom in the combined estimate isthen the sum of the individual degrees of freedom. Whenone uses an indenenc'ent estimate of the standard Jeviation,s , the test criterion recommended here for an outlier is as

follows:

x- X 1

S"

or:

X'n xn

,here:

total number of degrees of freedom.

25

, ' f I II

IThu. criticail values for T' and Tn" for the 5 percent

;,nd 1 '! ent significance levels are due to David (4) andare given in Table 5. In Table 5 the sub:icript v = dfIndicates the total number of degrees of freedom associatedwith the independent estimate of standard deviation cz andn indicates the number of observations in the sample understudy. We illustrate with an example on interlaborato-vtesting.

Lxa"', rr '. - >,.t c2'c or'ztcr , .", t. 7' - In an analysisof inteclaboratory test. procedures, data representingnormalities of sodium hydroxide solutions were determined bvtwL'lve different laboratories. In all rho standardizations,a 0.1 N sodium hydroxide solution was prepared by theStandard Methods Committee using carbon-dioxide-freedistilled water. Potassium acid phthalate (P.A.P.), obtainedfrom the National Bureau of Standards, was used is the teststandard.

Test data by the twelve laboratories are given inTable 6. The P.A.P. readings have hen coded to simplifythe calculations. The variances between the three readingswithin all laboratories were found to be homogeneous.A one-way classification in the analysis of variance vas firstanalyzed to determine if the variation in laboratory results(averages) was statistically signifi +ct. Th4s VaiLiunwas significant, and indicated a need for action, so testsfor outliers were then applied to isolate the particularlaboratories whose results gave rise to the significantvariation.

Table 7 shows that the variation between laboratoriesis highly significant. To test if this (very significant)variation is due to one (or perhaps two) laboratories thatobtained "outlying" results (that is, perhaps showing non-standard technique), we can test the laboratory averagesfor outliers. Prom the analysis of variance, we have anestimate of the variance of an individual reading as0.008793, based on 24 degrees of freedom. The estimatedstandard deviation of an individual measurement is

= 0.094 and the estimated standard deviation ofthe average of three readings is therefore

0.C94 /" = 0.054.

Since the estimate of within-laboratory variation isindependent of any difference between laboratories, we canuse the statistic T' above to test for outliers. An

l2

II1

examination of the deviations of the laboratory averay

VI. RECOMM IENDED CRITERIA FOR KNOWN STANDARD DEVIATION

Frequently the population standard deviation o maybe known accurately. In such cases, Table 9 may be used forsingle outliers and we illstrate with the foilowing example:

Exur-n?1 7 (o know.n) - Passage of the Echo I (Balloon)Satellite was recorded on star-plates when it was visible.Photographs were made by means of a camcra with shutterautomatically timed to obtain a series of points for theEcho path. Since the stars were also photographed at thesame times as the Satellite, all the pictures show star-trails and are thus called "star-plates."

1he x- and v-coordinate of each point on the Echopath are read from a photograph, using a stereo-comparator.

eliminate b4 as of the reader, the photograph is placedin one position and the coordinates are read; then thephotograph is rotated 180 deg. and the coordinates reread.The average of the two readings is taken as the final reading.Before any further calculations are made, the readings mustbe "screened" for gross reading or tabulation errors. ThisiS dorne bv examining the difference in the readings talenat the two positions of the photograph.

Table 10 records a sample of six reading- made at thet';o positions and the differences in t' ese readings. On th,third reading, the differences are ratier large. Has theoperator made an error in placing the cross-hair on the point?

For this example, an independent estimate of o isavailable since extensive tests on the stereo-comparator haveshown that the standard deviation in reader's error is about4,m. The standard deviation of the difference in tworeadings is therefore

4 z 7-+47 = 3 or 5.7 ,m

For the six readings (Table 10), the oean differencein the x-ccordinates is Lx = 3.5 and the mean differencein the y-coordinates is iy r 1.8. F or the questionablethird reading, we have

28

'4 33..5Tx 3.60

V 22 - 1.8 3.54y 5.7

From Table 9 we see that for n = 6, values of T' as

large as the calculated values would occur by chance less than1 percent of the time so that a significant reading errorseems to have been made on the third point.

A great number of points are read and automaticallytabulated on star-plates. Here we have chosen a verysmall sample of these points. In actual practice, thetabulat4.ons would probably be scanned quickly for very largeerrors such as tabulator errors; then some rule-of-thumbsuch a3 +3 standard deviations of reader's error might beused co scan for outliers due to operator error (Note 6).In o-her words, the data are probably too extensive to allowrepeated use of precise tests like those described above(especially for varying sample size), but this example doesillustrate the case where 7 is assumed known. If grossdisagreement is found ir the two readings of a coordinate,then fhe reading could be omitted or reread before furthercorputations are made.

,c'Ioe 6 - Note that the values of Table 9 vary between about1.4a and 3.S0oa.

VII. ADDITIONAL COMMIENTS

In the above, we have covered only that part ofscreening samples to detect outliers statistically. However,a large area remains after the decision has been reachedthat outliers are present in data. Once some of the sampleobservations are branded as "outliers," then a thoroughinvestigation should be initiated to determine the cause.In particular, one should look for gross errors, personalerrors of measurement, errors in calibration, etc. Ifreasons are found for aberrant observations, then one shouldact accordingly and perhaps scrutinize also the otherobservations. Finally, if one reaches the point that someobservations are to be discarded or treated in a specialmanner based solely on statistical judgment, then :*. mustbe decided what action should be taken in the further

29

analysis of the data. We do not propose to cover thisproblem here, since in many cases it will depend greatlyon the particular case in hand. However, we do remarkthat there could he the outright rejection of aberrantobservations once and for all on physical grounds (andpreferably not on statistical grounds generally) andonly the remaining observations would be used in furtheranalyses or in estimation problems. On the other hand,some may want to replace aberrant values with newlytaken observations and others may want to "Winsorize"the outliers, that is, replace them with the next closestvalues in the sample. Also, with outliers in a sample, somemay wish to use the median instead of the mean, and so on.Finally, we remark that perhaps a fair or appropriatepractice might be that of using truncated sample theory(IS) for cases of samples where we have "censored" orrejected some of the observations. We cannot go furtherinto these problems here. For additional reading onoutliers, see Refs (1, 3, 4, 14, 16, 17, 18).

Finally, a sample test criterion for non-normality,and hence possibly for outliers, not co:ered above is theVWilk-Shapiro W statistic for a sample of size n given by

[n/2] nW an~~ (X ~~ - x 2 / X', 2xi ] n-i+l (n'i+l ii

1' =31

u heren

SE x i P ,X1_- x2 _L X3 -< '"< Xn' i=l

[n/2] is the greatest integer in n/2, and the coefficientsa ni+l of the order statistics for n = 2(1)5n are given in

Shapiro, S. S. and M. B. Wilk, "An Analysis of VarianceTest for Non-Normality (Complete Samples)", BiometrikaVol. 52 (1965), pp 591-611, as is also a table ofpercentage points of W for n = 3(1)50.

The Wilk-Shapiro W statistic has been found to be

quite sensitive to departures from normality and may

compare most favorably with the 15- and b2 tests discussed

above. In addition, therefore, the W statistic may beused also as a test for outliers, or otherwise generalheterogeneity of sample values. Our significance testsgiven above have been selected and recommended since the)'generally "ocut" cut particular suspected outli*rs in thesample.

30

TABLE 1

Table of Critical Values for T(One-Sided Test of 1, or Tn )

When the Standard Deviation is Calculated from the Same Samples

No. Upper Upper Upper Upper Upper Upper

Obs. .1% Sig. .5% Sig. 1% Sig. 2.5% Sig. 5% Sig. 10% Sig.n Level Level Level Level Level Level

3 1.155 1.155 1.155 1.155 1.153 1.14F4 1.499 1.496 1.492 !.481 1.463 1.425S 1.780 1.764 1.749 ].715 1.672 3.6026 2011 1.973 1.944 1.887 1.822 1.729

2.201 2.139 2.097 2.020 1.938 1.8288 2.358 2.274 2.221 2.126 2.032 1.9099 2.492 2.387 2.323 2.215 2.110 1 .977

10 2. 606 2 .482 2 .410 2. 290 2 .176 2.036II 2.705 2 .564 2 485 2.355 .23 2 .08812 2.791 2 636 2.5S0 2.412 2.285 2.13413 2.967 2.699 2.607 2.462 2.331 2.17514 2.935 2 .755 2.659 2.507 2 371 2.21315 2.997 2. 806 2 .705 2 .549 2 .409 2.24716 3.052 2 .852 2.747 2 .585 2 .443 2.27917 3.103 2 .894 2 .785 2 .620 2 .475 2 30918 3.149 2.932 2.821 2.651 2.504 2.33519 3.191 2.968 2.854 2.681 2.532 2.361

L0 3.Z 3 .001 2 .884 2 .709 2 .557 2 .38521 3.266 3 .031 2 .912 2 .733 2 .580 2 .40822 3.300 3 .060 2.939 2 .758 2 .603 2 .42923 3.332 3.087 2.963 2.781 2.624 2.44824 3.362 3 .112 2 .987 2. 802 2.644 2 .467

25 3.389 3 .135 3.009 2. 82 2 663 2 .4626 3.415 3.157 3.029 2.841 2.681 2.502

27 3.440 3. 178 3.049 2. 859 2 .698 2 .51928 3.464 3 .199 3.068 2.876 2 .714 2 .53429 3.486 3 .218 3.085 2.893 2 .730 2 .549

30 3.507 3 .236 3. 103 2 .908 2 .745 2 .563

31 3.528 3.253 3.119 2.924 2.759 2.577

32 3.546 3 .270 3. 135 2 .938 2 .773 2 .59133 3.565 3 .286 3 150 2 .952 2 .786 2 .604

34 3.582 3.301 3.164 2.965 2.799 2.616

35 3.599 3.316 3.178 2.979 2.811 2.62836 3.616 3 330 3. 191 2.991 2 .823 2.639

37 3.631 3 .343 3.204 3.003 2 .835 2.650

38 3.646 3. 356 3.216 3.014 2 .846 2 661

39 3.660 3 .369 3.228 3.025 2 .857 2.67140 3.673 3. 381 3.240 3.036 2 .866 2 .682

31

A

TABLE I (CONTINUED)

Table of Critical Values for T(One-Sided Test of T1 or Tn)

When the Standard Deviation is Calcilated from the Same Samples

No. Upper Upper Upper tipper Upper UpperObs. .1% Sig. .5% Sig. 1% Sig. 2.5% Sig. 5% Sig. 10% Sig.n Level Level Level Level Level Level

41 3.687 3.393 3. 251 3.046 2.877 2 .69242 3.700 3.404 3.261 3.057 2.887 2.70043 3.712 3.415 3.271 3.067 2.896 2.71044 3.724 3.425 3.282 3.075 2.905 2,,1945 3.73f 3.435 3.292 3.085 2.914 2.72746 3.747 3.445 3.302 3.094 2.923 2.73647 3.757 3.455 3.310 3.103 2.931 2.74448 3.768 3.464 3.319 3 111 2.940 2.75349 3.779 3.474 3.329 3.120 2.948 2.76050 3.789 3.483 3.336 3.128 2.956 2.76851 3.798 3.491 3.345 3. 136 2.964 2 .77552 3.808 3.500 3.353 3.143 2.971 2.78353 3 816 3.507 3.361 3.151 2.978 2.79054 3.825 3.516 3.368 3.158 2.986 2.79855 3.834 3.524 3.376 3.166 2992 2.80456 3.482 3.531 3.383 3.172 3.000 2.81157 3.851 3.539 3.391 3.180 3.006 2.81858 3.858 3.546 3.397 3.186 3.013 2.92A59 3.867 3.553 3.405 3.193 3.019 2.83160 3.874 3.560 3.411 3.199 3.025 2.83761 3.882 3.566 3.418 3.205 3.032 2.84262 3.889 3.573 3.424 3.212 3.037 2.84963 3.896 3.579 3.430 3.218 3.044 2.85464 3.903 3.586 3.437 3.224 3.049 2.86065 3.9].0 3,592 3.442 3.230 3.055 2.86666 3.917 3.598 3.449 3.235 3.061 2.87167 3.923 3.605 3.454 3.241 3.066 2.87768 3.930 3.610 3.460 3.246 3.071 2.88369 3.936 3.617 3.466 3.252 3.076 2.88870 3.942 3.622 3.471 3.257 3.082 2.89371 3.948 3.627 3.476 3.262 3.087 2.89772 3.954 3.f33 3.482 3.267 3.092 2.90373 3.960 3.638 3.487 3.272 3.098 2.90874 3.965 3.643 3.492 3.278 3.102 2.91275 3.971 3.648 3.496 3.282 3.107 2.9176 3.977 3.654 3.502 3.287 3.111 2922

77 3.982 3.658 3.507 3.291 3.!!7 2.92778 3.987 3.663 3.511 3.297 3.121 2.93179 3.992 3.669 3.516 3. 301 3. 125 2 .93580 3.998 3.673 3.521 3.305 3.130 2.940

32

L __ __

TABLE I (CONTINIJED)

Table of Critical Values for T(One-Sided Test of T1 or T)

When the Standard Deviation is Calculated from the Same Samples

No. Upper Upper Upper Upper Upper UpperObs. .1% Sig. .5% Sig. 1% Sig. 2.5% Sig. 5% Sig. 10% Sig.n Level Level I.evel Level Level Level

81 4.002 3.677 3.525 3.309 3.134 2.945,32 4.007 3.682 3.529 3.315 3.139 2.94983 4.01.2 3.687 3.534 3.319 3.143 2.95384 4.017 3.691 3.539 3.323 3.147 2.95"85 4.021 3.695 3.543 3.327 3.1S1 2.96]86 4.026 3.699 3.547 3.331 3.155 2.966

4.031 3.704 3.551 3.335 3.160 2.9708 4 .035 3.708 3.555 7 3163 2.973

,89 4.039 3.712 3.559 3.343 3.16790 4.044 3.716 3.563 3.347 3.171 2.98191 4. 049 3 720 35 S67 3.350 3 .174 2. 98492 4 .053 3 725 3 570 3. 355 3. 179 2. 98993 4.057 3.728 3.575 3.358 3.182 2.99394 4.060 7 732 3. 579 3 .362 3 .186 .996(95 4.064 3. 736 3. 582 3.365 3. 189 3.00096 4.069 3. 739 3586 3.369 3.193 3 00397 4.073 3 .744 3.589 3 372 .196 (0 0t98 4.076 3.747 3.593 3.377 3.201 3.01199 4 .080 3. 750 3 .597 3 380 3 .204 3. 014100 ,..084 3. ?54 3.600 3.383 3. 207 3 017101 4.088 3. 757 3.603 3.386 3.210 3.021102 4.092 3.760 3.607 3.390 3.214 3.024103 4 .095 3. 765 3.610 3393 3. 217 3 .027104 4.098 3.768 3.614 3.397 3.220 3.030105 4 .102 3 771 3.617 3 .400 3. 224 3 .033106 4.105 3.774 3.620 3.403 3.227 3.037107 4.109 3.777 3.623 3.406 3.230 3.040108 4.112 3.780 3.626 3.409 3.233 3.043109 4.116 34.784 3.629 3.412 .3236 3.046110 4 .119 3 787 3 .632 3.415 3 .239 3 049ill 4.122 .790 3.636 3.418 3.242 3.052112 4.125 3.793 3.639 3.422 3.245 3.055113 4 129 3 796 3 64 2 3 424 3.218 3.058114 4 132 3 799 3 645 3.427 3.25) 3 061I15 4.135 3.802 3.647 3.430 3.254 3.064

116 4 .138 3. 805 3.650 3 433 3 257 3 .067117 4.141 3.908 3.653 3.435 3.259 3.0-0

33

TABLE I (CONTINUED)

Table of Critical Values for T(One-Sided Test of T1 or Tn)

When the Standard Deviation is Calculated from the Same Samples

No. Upper Upper Upper Upper Ulppcr UpperObs. .'.' Sig .5% Sig. 1% Sig. 2.5% Sig. S% Sig. 10% Sig.n Level Level Level Level Level Level

118 4.144 3.811 3.656 3.438 3.262 3.077119 4.146 3.814 3.659 3.441 3.265 3.07S120 4.150 3.817 3.662 3.444 3. 267 3.078121 4.153 3.819 3.66S 3.447 3.270 3.081122 4.156 3.822 3.667 3.450 3. 274 3.083123 4.159 3.824 3.670 3.452 3.276 3.086124 4. 161 3.827 3.672 3.455 3. 279 3. 089125 4.164 3.831 3.675 3.457 3.281 3.092126 4.166 3.833 3.677 3.460 3.284 3.095127 4.169 3.836 3.680 3.462 3.286 3.097128 4.173 3.838 3.683 3.465 3.289 3,100129 4.175 3.840 3.686 3.467 3.291 3.102130 4.178 3.843 3.688 3.470 3.294 3.104131 4.180 3.845 3.690 3.473 3. 296 3 .107132 4.183 3.848 3.693 3.475 3. 298 3. 109133 4.185 3.850 3.695 3.478 3.302 3.112134 4.188 3.853 3.697 3.480 3 .304 3 .114135 4.190 3.856 3.700 3.482 3.306 3.116136 4.193 3.858 3.702 3.484 3. 309 3 .119137 4.196 3.860 3.704 3.487 3. 311 3 .122138 4.198 3.86S 3.707 3.489 3.313 3.124139 4.200 3.865 3.710 3.491 3.315 3.1'6140 4.203 3.867 3.712 3.493 3.318 3.129141 4.205 3.869 3.714 3.497 3.320 3.131142 4.207 3.871 3.716 3.499 3.322 3.133143 4.209 3.874 3.719 3.501 3.324 3.135144 4.212 3.876 3.721 3.503 3.326 3.138145 4.214 3.879 3.723 3.505 3. 328 3.140146 4.216 3.881 3.725 3.507 3 331 142147 4 219 3.883 3.727 3.509 3. 334 . 4

34

LI

TABLE DIXON CRITERIA FOR TESTING OF EXTREME OBSERVATION

(SINGLE SAMPLE)a

Signi.ficance Level

n Criterion 10 1

3 r 10 = (x 2 - X1)/(x n xj) if smallest 0.886 0.941 0.98P

4 value is suspected; 0.679 0.765 0.8895 = (x n -n )/(x n - xI) if 0.557 0.642 0.780

6 largest value is suspected. 0.482 0.560 0.6987 0.434 0.507 0.637

8 rl] = (x2 - xl)/(xn -1 x1 ) if 0.479 0.554 0.68-

9 smallest value is suspected; 0.441 0.512 0.63510 = (x - xn M)/(x n x 2 ) if 0.409 0.477 0.597

largest value is suspected.

11 r 2 1 = (x 3 - xl)/(xn 1 - x ) if 0.517 0.576 0.679

12 smallest value is suspected; 0.490 U.46 0 .64213 = (xn - Xn2 )/(xn - x2 ) if 0.467 0.521 0.615

largest value is suspected.

14 r2 2 = (x3 xl)/(Xn 2 - x1) if 0.492 0.546 0.641

15 smallest value is suspected; 0.472 0.525 0.61616 = (x n - x n ,)/(x n x 3 ) if 0.454 0.507 0.595

largest value is suspected. 0.43I 0.490 0.57"18 0.424 0.475 0.561

19 0.412 0.462 0.54?20 0.401 0.450 0.535S1 0.391 0.440 0.524

22 0.382 0.430 0.51423 0,374 0.421 0.505

24 0.367 0.413 0.49,25 0. 360 0.,106 0.48Q(

ax ... x . (See Ref(7), Appendix.)

35

'-l ' - l ~ l i l- l -- - -•

TABLE 3

CRITICAL VALUES FOR w/s (RATIO OF RANGE TO SAMPLE

STANDARD DEVIATION)a

Number of 5 Percent I Percent 0.5 PercentObservations Significance Significance Significance

n Level Level Level

3 2.00 2.00 2.004 2.43 2.44 2 .455 2.75 2.80 2.816 3.01 3.10 3.127 3.22 3.34 3.378 3.40 3.54 3 589 3 .55 3.72 3.77

10 3.68 3.88 3.9411 3.80 4.01 4.0812 3.91 4.13 4.2113 4.00 4.24 4.3214 4.09 4.34 4.4315 4.17 4.43 4.5316 4.24 4.51 4.6217 4.31 4.59 4.6918 4.38 4.66 4.7719 4.43 4.73 4.8420 4.49 4.79 4.9130 4.89 5. 2S 5. 3940 5.15 5S4 5.6950 5 35 5.77 S.9160 5 50 S. 93 6.0980 5.73 6.18 6. 35

100 S.90 6.36 6.54150 6.18 6.64 6.84200 6.38 6.85 7.03500 6.94 7.42 7.601000 7.33 7.80 7.99

aSee Ref (S), where:

. x Xn -X XI I-- X 2 -

TABLE 4 *

TABLE OF CRITICAL VALUES FOR S2_ 1 ,n/S OR S 21, /S FOR

SIMULTANEOUSLY TESTING THE TWO LARGEST OR TWO SMALLESi

0 B S IL P VAT I': S

No. of Lower Lower Lower Lower Lower LowerObs. .1% Sig. .5% Sig. 1% Sig. 2.5% Sig. 5% Sig. 10% Sig.n Level Level Level Level Level Level

1

4 .0000 .0000 .0000 .0002 .0008 .0031S .0003 .0018 .0035 .0090 .0183 .03766 .0039 .0116 .0186 .0349 0564 .09207 .0135 .0308 .0440 .0708 .1020 .14798 .0290 0563 .0750 .1101 .1478 .19949 .0489 .0851 .1082 .1492 .1909 .2154

10 .0714 .1150 .1414 .1864 .2305 .286311 .0953 .1448 .1736 .2213 .2667 .321212 .1198 .1738 .2043 .2537 .2996 .355213 .1441 .2016 .2333 .283b .3295 .384314 .1680 .2280 .2605 .3112 .3568 .4106is .1912 .2330 .2859 .3367 .3818 .434516 .2136 2767 .3098 .3603 .4048 .4562-17 .2350 .2990 .3321 .3822 .4259 .476118 .2556 .3200 .3530 .4025 .4455 .494419 2752 3398 .3725 .4214 .4636 .51]320 .2939 .3585 .3909 .4391 .4804 527021 .3118 .3761 .4082 .4556 .4961 .541522 .3288 .3927 .4245 .4711 .5107 555023 .3450 .4085 .4398 .4857 5244 .567724 .36cj5 .4234 .4543 .4994 .5373 579525 .3752 .4376 .4680 .5123 $495 5906

n nS 2 X X) ( . x 2 X E X 1 .)

n i=n

S' 2 F (x x ,2)2 X1, 2 E xi,2 i=2 n-2 i=2

n-2 n n-2

S2 : z (x. x )2 xn- , i1 n-l,n 2 n-1, n n_ j x

*These significance levels are taken from Table Ii, fef. (12).

An observed ratio less than the appropriate critical ratio

in this table calls for rejection of the null hypothesis.

37

. ~ . . . . . . . . _ _ __ _ _ _ _=

TABLE 4 (CONTINUED) ISR 2 2/S2FRTABLL OF CRITICAL VALUES FOR S_ /2 OR FOR

n-1,n I1,2SIMULTANEOUSLY TESTING THE TWO LARGEST OR TWO SMLLIST OBSERVATIONS

No. of Lower Lower Lower Lower Lower LowerObs. .1% Sig. .5% Sig. 1% Sig. 2.5% Sig. 5% Sig. 10% Sig.

n Level Level Level Level Level Level

26 .3893 .4510 .4810 .245 .5609 .601127 .4027 .4638 .4933 5360 5717 .611028 .4156 4759 .5050 5470 .5819 .620329 .4279 4875 .5162 .5574 5916 .629230 .4397 4985 .5768 5672 6008 637531 .4510 5091 .5369 5766 .6095 tj45532 .4618 .5192 5465 5856 .6178 .653033 .4722 5288 5557 5941 6257 .660234 .4821 .5381 5646 .6023 .6333 .667135 .4917 5469 5730 .6101 .6405 636 5009 5554 .5811 .6175 674 .6800

.5098 5636 5889 .6247 654 .686038 .5184 .5714 5963 .6316 .6604 .6917.9 5266 .5789 .6035 .6382 .6665 .697240 5345 .5862 .6104 .0445 .6724 .702541 .5422 .5932 6170 6506 .6780 707042 5496 5999 .6234 .6565 .6834 .712543 5568 .6064 .6296 .6621 .6886 717244 .5637 .6127 .6355 6t76 .6136 72145 5704 .6188 .6412 .6728 o985 .26140 5768 .6246 .6468 6779 .7032 .730447 5831 .6303 .6521 .6828 .7077 .734548 5892 .6358 .6573 .6876 .7120 .738449 .5951 .6411 .6623 .(921 .7163 .742250 .6008 .6462 .6672 .6966 .7203 .745951 .6063 6512 .6719 7009 .7243 749552 .6117 6560 .6765 .7051 .7281 .752953 .6169 .6607 .6809 .7091 .7319 .756354 .0220 .6653 .6852 .7130 .7355 .759555 .6269 .6697 .6894 .7168 .7390 .762756 6317 .6740 .6934 7205 .7424 .765857 .6364 .6782 .6974 .7241 .7456 .768758 .6410 .6823 .7012 .7276 .7489 .771659 .6454 .6862 .7049 .7310 .7520 7744(10 .6497 .69Yi .7086 .7343 .7550 777261 .6539 .6938 .7121 .7375 .7580 779862 .6580 .6975 .7155 .7406 .7608 782463 .6620 .7010 .7189 .7437 .7636 .785064 .6658 .7045 .7221 .7467 .1661 7874

6S .6696 7079 .7253 .7496 .7690 789866 .6733 .7112 .7284 .7524 .7716 ,792]

67 .6770 .7144 .7314 .7551 .7741 .7944

38

L _ _ __--_

TABLE 4 (CONTINUII))

TABLE 01: CRITICAL VALUES FOR S2 /s2 OR S- /S2 [ORn-1,n ,2

SIMULTANOUSLY TISTING THE TWO LARG(LST OR TWO SHALLST

OBSERVATIONS

No. of Lower Lower Lower Lower Lowe I.owcrOhs. .1% Sig. .5% Sig. 1% Sig. 2.5% Sig. 5% Sig. 10% Sig.

n Level Level Level Level Level Level

68 .6805 .7175 .7344 .7578 7766 79(,()

69 .0839 .7206 .7373 .7604 .7790 .798870 .6873 .7236 .740] .7630 .7813 .800971 .6906 .7265 .7429 .7655 .7836 .8030

.6938 .7294 .7455 .7679 .7859 .805.0S.69 0 .7322 .7482 7703 .7881 .807()

74 .7000 .7349 .7507 .7727 .7902 .808975 .7031 7376 7532 .7749 .7923 .81087o .7000 .7402 .7557 .7772 .7944 .812777 .7089 .7427 .7581 .7794 .7964 .81457 .7117 74S3 .7605 .7815 .7983 .8162

.7145 .7477 .7628 .7836 .8002 .818F,s0 7172 .7501 .7650, .7856 .8021 .819"1 .7!99 .7525 .7672 .787b .8010 8213

82 7225 .7548 .7694 .7896 .8058 .823083 .7250 7570 7715 7915 8075 8215

84 .7275 .7592 .7736 .7934 .8093 826185 .7300 .7611 .7756 q7953 .109 .827()

86 7324 7635 7776 7971 8126 .2918q7 7348 7656 .7790 .7989 8142 .300)88 7371 .7677 .7815 .8006 .8158 832 189 .7394 .7697 .7834 .8023 .8174 .8335

90 7416 7717 .7853 .8040 .8190 8 349

91 7438 7736 .7871 .8057 .8205 36292 .7459 *7755 .7889 8073 8220 .837o

93 7481 7774 .7900 .8089 .8234 .838994 .7501 .7792 .7923 .8104 .8248 .840295 .7522 .7810 .7940 .8120 .82o3 .8414

96 .7542 .7828 .7957 .8135 .8270 8127

97 .7562 .7845 .7973 .8149 .8290 .8139

98 .7581 .7862 .7989 .8164 .8303 .8451

99 .7600 .7879 '8005 .8178 .8316 .8433

100 .7619 .7896 .8020 .8192 .8329 .8475101 .7637 .7912 .8036 .8206 .8342 .8iso102 .765 5 -928 .8051 .8220 8354 8,97107 7673 7944 8065 .8233 8367 7 .8104 769 .7959 .8050 .8246 S379 8519105 7708 7974 8094 82 S.9) p391 .S"%0106 7725 .7989 8108 .8272 .8402 .8541

39

[.

I

TABLE 4 (CON!INUED)TABLE OF CRITICAL VALUI S FOp S2-l,92 OR S2 ,/2 FOP,

n - I .-

SIMULTANFOIISLY TES' IN '1I1E TWO LARGEST 0R TWO SMALILST

B S F R VAT (1 N S

No. of lower Lower l.ower Lower Lower I., Ohs, .1% Sig. .51 Sig. 1% Sig. 2.5% Sig. S%, Sig. 10c Sig.

n Level Leve 1 Leve1 lox-cl Level Level

10- 7742 .8004 .8122 8284 8414 .8551108 7758 .8018 .8136 .8297 .8425 .8503109 .7774 .8033 .8149 8309 .8436 8571110 .790 8047 .8162 .8321 .8447 8581111 806 .80o1 .8175 .8333 .8458 .8591112 . 8074 .8188 .834,1 84 0 ) .8(0113 .7837 .8088 .8200 .8356 .84'79 .8610114 7852 .8101 .8213 8367 .8489 8019115 786o 8114 8225 8378 .8500110 7881 8127 8237 .8389 .8510 8o37117 .7895 .8139 .8249 .8400 8519 80,,0118 .7909 .8152 .8261 .8410 .8529 .8o55119 9"923 8 "8272 .8421 .8539 .)04120 -937 .8170 .8284 .84 31 .8517121 795 .8188 .8295 .8441 8557 8081122 97904 .8200 .8300 8451 8z$6 7 8)89125 "7977 .8211 .. 17 8461 8576 809712. 7990 .8223 .8327 .8471 .8585 8705

.800 3 823a .8338 .8480 8593 8713126 .8010 .8245 .8348 .8490 .8002 .8721127 8028 .8250 8359 8499 8o1.1 .8729

128 .8041 .8267 .8369 8508 .8619 .8737129 .8053 .8278 .8379 .8517 .8o27 8.4130 .8065 .8288 .8389 .8526 .863o .8752131 .8077 .8299 .8398 .8535 .8044 .87591 32 .8088 .8309 .8408 .8544 .8652 .87,6133 .8100 .8319 .8418 .8553 .8660 .8773134 .8111 .8329 .8427 .8561 .8668 8780

135 .8122 .8339 .8436 8570 .8675 .8787

] 1 .8134 .8349 .8 415 .8578 .8 o ,8 879 ,137 .8145 .8358 .8454 .8586 .8090 880113s .8155 .8368 .8463 .8594 .8698 .8808139 .8166 .8377 .8472 .8602 .8705 . 811.140 .8176 .8387 .8481 .8610 .8712 .. 211a1 .8187 .8396 .8489 .8618 8-720 .8827142 .819- 8405 .8498 Jo25 8727 .8834143 .8207 .8414 .8506 .8633 .873.1 .8840144 8218 8423 .8515 8041 .8741 .8846145 .8227 .8431 .8523 8t48 8747 8853146 .8237 .8440 .8531 .8655 .8754 .59

147 .82i" .8449 8539 .8663 .8701 .8865148 .825O .8457 8547 .8670 .8767 .8871149 .1266 .8165 .8555 .8677 .8774 .8877

40

CR1ITICAL VALULS FOR 'I' WHlIN STANI)AR1) 1) 1 .'IlfN I 1 N

x -x X-XI- n-

nS S

. percentage point

10 2,78 3.10 3.32 3.48 3. 2 3.73 3 . 0.90 4.o411 2,72 3.02 3.24 3.39 3.5 3.03 3.72 3,79 3.9312 2..7 32 3.39-234.7 3..31 .63 2.92 3 .12 27 3.38 3.48 S7 3oI .

14 2.60 2.88 07 3.22 3.33 3.43 3 .51 358 5.-0

15 257 2.84 3.03 3.17 329 . .38 34 3.5]0 54 2.81 3.00 14 3.25 3.34 .42 .49 . 017 22 2. .7 .11 3.22 3.31 3.3 .15is 2 50 .77 295 08 3.19 3 28 3,35 3 . ,19 2.49 2.75 2 .93 306 3.1 3.25 3.33 -.9 s

20 2.47 2.91 3.0,1 11 3 .2. 3.30 .3- 3. 44 2.42 08 2.84 2.97 3.07 3.1 3. 23 .29 .

30 2.38 2.o2 2.791 2 .1 3.C,1 30S 3,15 3, 3.40 24 2 .5 273 2.85 2. 94 302 .o 3i 3 3 2

..... I .7 -.8 , ~120 2925 2.60 1 u I.0 2 2S 2 48 .2 . ,2 73 8- 2 89 2 95 3.00 3.08

_4 2.__7 j .68 .76 2 _.83 2 8h 2.93 3.01Spercentage points

10 2.O1 2.27 2.46 2.60 12.8122.89 2.96 .01 .98 224 2.42 2.56 2.67 2: 7o 84 91 .

12 1.96 2.21 2.39 2 .52 2.63 2.7 2.80 87 2.9813 . .1 .94 l) . 1.9 0 . O 2. .-, G 2 .69 2 .a C76 83 2 ., 9 414 1.93 2.17 2.34 2. 47 2.57 2.66 ?.74 2.80 2.91

15 1.91 2.15 2.32 2.45 2 . 2. 4 2 .71 -,77 7 .16 1 90 2 14 2.31 2.43 2.S3 2.62 2.69 . 2.817 1 .89 "13 2.29 2 2 2.52 2.60 73 2.418 1.8R 2.11 2 . 2.40 2.. s 5 5 2 .05 2 . 7 1 .219 1 87 2.11 2.27 2 .39 2.49 2.57 2.4 .70 2.,0

20 i .87 10 26 .R :,47 ? S6 2.()3 2 27824 1.84 2.07 2.23 2.34 2.44 2.52 28 2 .(,4 7430 1.82 2.04 2.20 2 31 2.40 2.48 2.5 1 2.60 2 .64, 1.80 2.02 2.17 2 8 2 ";7 2.44 2.So 2.So 2.05

610 1 .78 1 .99 2 .14 2.25S 2 .33 2 .4I1 247 2.52120 1 . 76 1 . 96 2 . 11 2 . 2 30 2 . 7 243 ..3 2 7

1 74 1 94 2 .08 2.18 2.27 2.33 2.37 ..14 j2.5

hc pcrcentzigc points a 1 I re r cproduced r I . ( 1.

41

4i4

TABLE 6

STANDARDIZATION OF SODIUM HYDROXIDL SOLUIIONS AS DETERMINEljBY PLANT LABORATORILS

Standard Used: Potassium Acid Phthalate jP.A.P.)Deviation

(P.A.P. - of Average0.096000) from Grand

Laboratory X 103 Sums Averages Average_1 1.893

1.9721.876 5./41 1.914 +0.043

2 2.0461.851

1.949 5.846 1.949 +0.0831 .874 - '

1.7921_829 5.495 1.832 -0.039

148 *0.013

1.998

1.98

1.9830 4 97 + 7

2.066 6.038 2.013 +'.1422.020

I .903 6.134 2.045 +0.1741.8831.855 5.569 1.8S6 -0.015

10 0._______I 0.777 2.234 0.74S -1.126

11 2.064

1 .891 5.749 1.916 +0.04512 2 17

S '.102 6.980 2.327 +0.456

Grand Sum .... 67.350 ______ __

Grand Axvcrage . . .. _ _____ 1.871 ______

42

L)r

-444cI -4

0 t) m Go'~4 Ln - '-

(I) V) 0. CO) tII '- A CDV 4 CD

a ) C=) CDA-

oo

C4 cz.I C7.r-- Nle cu 0 CD~ 00 T)~

- -- I40 14(

>--4

~1~

TABLE 9

CRITICAL VALUES OF T1' AND T' WHEN THE POPULATION STANDARD

DEVIATION o IS KNOWNa

Number of 5 Percent 1 Percent 0.5 PercentObservations Significance Significance Significance

n Leve 1 Level Level

2 1.39 1,82 1.993 1. 74 2.22 2.404 1.94 2.43 2 625 2.08 2.57 2.766 2.18 2.68 2.877 2.27 2.76 2.95

8 2.33 2.83 3.029 2.39 2.88 3.07

10 2.44 2.93 3.1211 2.48 2.97 3.161 2.52 3.01 3.20

13 2.56 3.04 3.2314 2.59 3.07 3.2615 2.62 3.10 3.2916 2.64 3.12 3.3117 2.67 3.15 3.3318 2.69 3.17 3.3619 2.71 3.19 3.3820 2.73 3. 21 3.39

21 2.75 3.22 3.4122 2.77 3.24 3.4223 2.78 3.26 3.4424 2.80 3.27 3.4525 2.81 3.28 3.46

X . X 2 X 3 . . . X n

T'. x ' = (xn - )/'o

aThis table is taken from Ref (10).

44

TABLE 10

MEASUREMENTS, lim

x-Coordinate y-Coordinate

Position on

Position 1 1 + 180 I x Position 1 1le8 Ay

deg deg

-53011 -53004 -7 70263 70258 +5

-3811' -38103 -9 -39729 -39723 -6

2804 - 2828 +24 81162 81140 +22

18473 18467 +6 41477 41485 -8

25507 25497 +10 1082 1076 +6

87736 87739 -3 -7442 -7A34 -8

45

TABLE *11

Critical Values for Lk - 0.01ka

n/k 1 2** 3 4 5 6 7 8 9 10

3 .000 .000

4 .011 .010 .000 .000

5 .045 .044 .004 .004

6 .091 .093 .021 .019 .002

7 .148 .145 .047 .044 .010

8 .202 .195 .076 .075 .028 .008

9 .235 .241 .112 .108 .048 .018

10 .280 .283 142 .142 .07U .032 .012

11 .327 .321 .178 .174 .098 .052 .026

12 .371 .355 .208 .204 .120 .070 .038 .019

13 .400 .386 .233 .233 .147 .094 .056 .033

14 .424 .414 .267 .261 .172 .113 .072 .406 .027

15 .450 .440 .294 .286 .194 .132 .090 .057 .037

16 .473 .463 .311 .310 .219 .15] .108 .072 .049 .030

17 .480 .485 .338 .332 .237 .171 .126 .091 .064 .044

18 .502 .504 .358 .353 .260 192 .140 .104 .076 .053 .036

19 .508 .522 .366 .373 .272 .201 .154 .118 .088 .064 .046

0 .533 .539 .387 .391 .300 .231 .175 .136 .104 .078 .058 .042

,S 603 .468 .377 .308 .246 .204 .168 .144 .112 .092

30 . SO .526 .434 .369 312 268 .229 .19( 166 .142

5 .690 .574 .484 .418 364 321 282 .250 220 .194

40 722 .608 522 .460 .408 364 324 202 .o2 .234

45 745 .636 558 .498 444 399 361 328 .296 .270

50 .768 .668 .592 531 .483 .438 .400 .368 .336 .308

* From Grubbs (1950, Table I).

** Fron Grubbs (1950, Table V).

(This table is taken from Tietjen and Moore, Reference 19)

46

TABLE 11

Critical Values for Lk 0.025

n/ 1 1* 2 2** 3 4 5 6 7 8 9 10

3 .001 .001 .000 .000

4 .025 .025 .000 .000

5 .084 .081 .011 .009

6 .146 .145 .034 .035 .005

7 .209 .207 .076 .07] .0218 .262 262 .11S .110 .045 .013

9 .308 .310 .150 .i49 .073 .030

10 .350 .353 .188 .187 .100 .052 .023

11 .366 .390 .225 .221 .129 .074 .040

12 .440 .423 .268 .254 .162 .096 .057 .03113 .462 .453 .292 .284 .184 122 .077 .04714 .493 .479 .317 .311 .214 .145 .098 .063 .03815 .498 .503 .341 .337 .239 .167 .111 .078 .051

16 537 .525 .372 .360 .261 .185 .137 .096 .065 .045

17 .552 .544 .388 .382 .282 .208 !156 .117 .082 .058

18 .570 .562 .406 .403 .299 .226 .171 .129 .095 .068 .04819 ..73 .579 .416 .421 .311 .243 .189 .145 .108 .080 .059

20 595 .594 .442 .439 .341 .265 .209 .165 .128 .098 .C.73 .05425 .656 .654 516 .417 .342 .282 .233 .192 .159 .132

30 .699 .568 .479 .408 .352 .302 .261 226 .193 .165

35 .732 .612 .527 .435 .398 .348 .308 .274 .242 .213

40 .755 .641 .561 .491 .433 .387 .348 .314 .283 .257

45 .773 .667 .592 .529 .473 .430 .391 .356 .325 .295

50 .796 .698 .622 .559 .510 .466 .428 .392 .363 .334

*From Grubbs (1950, 'able 1).**Fiom Grubbs (1950, Table V).

47

TABLE 11

Critical Values for Lk = 0.05

n/ 1 1j 2 2** 3 4 5 6 7 8 9 10

3 .003 .003

4 .051 .049 .001 .001

5 .125 .127 .018 .018

0 .203 .203 .055 .057 .010

7 .273 .270 .106 .102 .032

8 326 .326 .146 .148 .064 .022

9 .372 .374 .194 .191 .099 .045

10 .418 .415 .233 .230 .129 .070 .034

11 .454 .451 .270 .267 .162 .098 .054

12 .489 .482 .305 .300 .196 .125 .076 .042

13 .517 S O .337 .330 .224 .150 .098 .060

14 540 .534 .363 .357 .250 .174 .122 .079 .050

15 .556 556 .387 .382 .276 .197 .140 .097 .066

16 .575 .576 .410 .405 .300 .219 .159 .115 .082 .055

17 .594 .593 .427 .426 .322 .240 .181 .136 .100 .072

18 .608 .610 .447 .446 .337 .259 .200 .154 .116 .086 .062

19 .624 .624 .462 .464 .354 .277 .209 .168 .130 .099 .074

20 .639 .638 .484 .480 .377 .2S9 .238 .188 .150 .115 .088 .066

25 .696 .692 .SSO .450 .374 .312 .262 .222 .184 .154 .126

30 .730 .599 .506 .434 .376 .327 .283 .245 .212 .183

35 .762 .642 .554 .482 .424 .376 .334 .297 .264 .235

40 .784 .672 .588 .523 .468 .421 .378 .342 .310 .280

45 .802 .696 .618 .556 .502 .456 .417 .382 .350 .320

30 .820 .722 .04b .588 .535 .490 .450 .414 .383 .356

*From Grubbs (1950, Fable I).**From, Grubbs (1950, Table V).

48

TABLE 11

Critical Values for Lk = 0.10k

n/ 1* 2** 3 4 5 6 7 8 9 10

3 .011 .011

4 .098 .098 .003 .003

5 .200 .199 .038 .038

6 .280 .283 .091 .092 .020

7 .348 .350 .148 .148 .056

8 .404 .405 200 .199 .095 .038

9 .448 .450 .248 .245 .134 .068

10 .490 .488 .287 .286 .170 .098 .051

11 .526 .520 .326 .323 .208 .128 .074

12 .555 .548 .361 .355 .240 .159 .103 .062

13 .578 .S73 .388 .384 .270 .186 .126 .082

14 .600 594 .416 .411 .298 .212 .150 .104 .068

15 .611 .613 .436 .435 .3Z2 .236 .172 .124 .086

16 .631 .631 .458 .456 .342 .260 .194 .144 .104 .073

17 .b48 .646 .478 .476 .364 .282 .216 .165 .125 .092

18 .661 .660 .496 .494 .384 .302 .236 .184 .142 .108 .080

19 .676 .673 .510 .511 .398 .316 .251 ,199 .158 .124 .094

20 .688 .685 .530 .527 .420 .339 .273 .220 .176 .140 .110 .085

25 .732 .732 .588 .489 .412 .350 .296 .251 .213 .180 .152

30 .766 .637 .523 .472 .411 .359 .316 .276 .240 .210

35 .792 .673 .586 .516 .458 .410 .365 .328 .294 .262

40 .812 .702 .622 .554 .499 .451 .408 .372 .338 .307

45 .826 .724 .648 .586 .533 .488 .447 .410 .378 .348

50 .840 .744 .673 .614 .562 .518 .477 .442 .410 .380

*From Grubbs (1950, Table I)**From Grubbs (1950, Table V).

49

L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

TABLE 12

Critical Values for Ek,a = 0.01

n/k 1 2 3 4 5 6 7 8 9 10

3 .000

4 .004 .000

S .02 002

6 .068 U12 .001

7 .110 .028 .006

8 .156 .050 .014 .004

9 .197 .078 .026 .009

10 .235 .101 .018 .006

11 .274 .134 .064 .030 .012

12 .311 .159 .083 .042 .020 .008

13 .337 .181 .103 056 .03J .014

14 .374 .207 .123 .072 .042 .022 .012

15 .404 .238 .146 .090 .054 .032 .018

16 .42? .263 .166 .107 .068 .040 .024 .014

17 .440 .290 .188 .122 .079 .052 .032 .018

18 .459 .306 .206 .141 .094 .062 .041 .026 .014

19 .484 .323 .219 .156 .108 .074 .OSO .032 .020

20 .499 .339 .236 .170 .121 .086 .058 .040 .026 .017

25 .571 .418 .320 .245 .188 .146 .110 .087 .066 .050

30 .624 .482 .386 .308 .250 .204 .166 .132 .108 .087

35 .669 .533 .435 .364 .299 .252 .211 .177 .149 .124

40 .704 574 .480 408 .347 .298 .258 .220 .190 .164

45 .728 .607 .518 .446 .386 .336 .294 .258 .228 .200

SO .748 .636 .550 .482 .424 .376 .334 .297 .264 .235

(This table is taken from Tietjen and Moore, Reference 19)

so

TABLE 12

Critical Values for Ek 0.05

kn/ 1 1* 2 3 4 5 6 7 8 9

3 .001 .001

4 .C25 .025 .001

5 .081 .081 .010

6 .146 .145 .034 .004

7 .208 .207 .065 .016

8 .265 .262 .099 .034 .010

9 .314 .310 .137 .057 .021

10 .356 .352 .172 .083 .037 .014

11 .386 .390 .204 .107 .055 .026

12 .424 .423 .234 .133 .073 .039 .018

13 .455 .453 .262 .156 .092 .053 .028

14 .484 .479 .293 .179 .112 .068 .039 .021

15 .509 .503 .317 .206 .134 .084 .052 .030

16 .526 .525 .340 .227 .153 .102 .067 .041 .024

17 .544 .544 .362 .248 .170 .116 .078 .050 .032

18 .562 .562 .382 .267 .187 .132 .091 .062 .041 .026

19 .581 .579 .398 .287 .203 .146 .105 .074 .050 .033

20 .597 .594 .416 .302 .221 .163 .119 .085 .059 .041 .028

25 .652 .654 .493 .381 .298 .236 .186 .146 .114 .089 .068

30 .698 .549 .443 .364 .298 .246 .203 .166 .137 .112

35 .732 .596 .495 .417 .351 .298 .254 .214 .181 .154

40 .758 .629 .534 .458 .395 .343 .297 .259 .223 .195

45 .778 .658 .567 .492 .433 .381 .337 .299 .263 .233

50 .797 .684 .S99 .529 .468 .417 .373 .334 .299 .268

*From Grubbs, Ref. (11), 1950.

51

h. =_-~ -. . . . . . ___

TABLE 12

Critical Values for Ek - 0.10

n/ 1* 2 3 4 5 6 7 8 9 10

3 .003 .003

4 .350 .049 .002

5 .127 .127 .022

6 .204 .203 .056 .009

7 .268 .270 .094 .027

8 .328 .326 .137 .053 .016

9 .377 .374 .175 .080 .032

10 .420 .415 .214 .108 .052 .022

11 .449 .451 .250 .138 .073 .036

12 .485 .482 .278 .162 .094 .052 .026

13 .510 .510 .309 .189 .116 .068 .038

14 .538 .534 .337 .216 .138 .086 .052 .029

15 .558 .556 .360 .240 .160 .105 .067 .040

16 .578 .576 .384 .263 .182 .122 .082 .053 .032

17 .594 .593 .406 .284 .198 .140 .095 .064 .042

18 .610 .610 .424 .304 .217 .156 .110 .076 .051 .034

19 .629 .624 .442 .322 .234 .172 .124 .089 .062 .042

20 .644 .638 .460 .338 .252 188 .138 .102 .072 .051 .035

25 .693 .692 .528 .417 .331 .264 .210 .168 .132 .103 .080

30 .730 .582 .475 .391 .325 .270 .224 .186 .154 .126

35 .763 .624 .523 .443 .379 .324 .276 .236 .202 .172

40 .784 .657 .562 .486 .422 .367 .320 .278 .243 .212

45 .803 .684 593 522 .459 .406 .360 .320 .284 .252

50 .820 .708 .622 552 .492 .440 .396 .3S5 .319 .287

*From Grubbs, Ref. (1i), 1950.

52

REFERENCES

(1) AnsQmbe, F. J., ,R-jection of ottlier"',Technometrics, Vol 2, No. 2, 1960, pp 123-147.

(2) Chauvenet, W., A Manual of Spherical and Prcctica&Astronomy, Vol 2, Fifth Edition, 1876.

(3) Chew, Victor, "Tests for the Rejection of OutlyingObservations", RCA Systems Analysis TechnicalMemorandum No. 64-7, 31 Dec 1964, Patrick Air ForceBase, Fla.

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53

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(15) Kudo, A., "Oi the Testing of Outlying Observations",$a;~h, 7he hndian ,Jcurral of 'tatistics, Vol 17,Part i, June 1956, pp 67-76.

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(17) Sarhan, A. F., and Greenberg, B. C., Edicors,Cortpibuicns to Crder Statistics , John Wiley andSons, Inc. , Nev Ycrk, 1962.

(18) Thompson, N. R., "On a Criterion for the Rejectionof Observations and the Distribution of the Ratio ofthe Deviation to the Sample Standard Deviation"The AnnaZs of "atheratical Statistics, Vol 6, 1935,pp 214-219.

(19) Tietjen, G. I.. and Moore, R. H., "Some Grubbs-TypeStatistics for the Detection of Several Outliers',Technometrics, Vol 14, No. 3, August 1972, pp 583-597.

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